#449470

order	512
nilpotency class	3
isoclinism	isoclinic class of #418851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2679
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15947
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13)(4,18,21,9)(5,10)(6,25)(7,12)(8,16)(11,28)(14,40,44,58)(15,41)(17,38)(19,33,51,47)(20,34,54,48)(22,55)(23,39)(24,37)(26,31)(27,36)(29,52,32,43)(30,46)(35,45)(42,62,63,59)(49,56,64,57)(50,61,53,60);(1,3)(2,8)(4,14)(5,22)(6,26)(7,17)(9,32)(10,35)(11,38)(12,31)(13,25)(15,23)(16,28)(18,29)(19,49)(20,52)(21,44)(24,46)(27,45)(30,36)(33,59)(34,40)(37,41)(39,55)(42,50)(43,54)(47,62)(48,58)(51,64)(53,63)(56,61)(57,60);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,29)(10,33)(11,34)(12,18)(13,40)(15,42)(16,43)(17,44)(22,49)(23,50)(24,51)(25,48)(26,52)(27,53)(28,54)(30,56)(31,32)(35,62)(36,60)(37,47)(38,58)(39,61)(41,59)(45,63)(46,64)(55,57);(1,5)(2,10)(3,15)(4,19)(6,27)(7,24)(8,30)(9,33)(11,39)(12,37)(13,41)(14,42)(16,46)(17,45)(18,47)(20,53)(21,51)(22,26)(23,28)(25,36)(29,56)(31,55)(32,57)(34,61)(35,38)(40,59)(43,64)(44,63)(48,60)(49,52)(50,54)(58,62)

Polycyclic presentation:

2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356666);